To find the amount of work required to empty the trough, we can break down the problem into two parts:
1. The work required to raise the water to the top of the trough.
2. The work required to lift the water from the top of the trough to the outlet.
Let's start with the first part:
1. The work required to raise the water to the top of the trough:
- The mass of the water in the trough can be calculated by multiplying the volume of the trough by the density of water. The volume of the trough can be found by multiplying the length, width, and depth of the trough.
- In this case, the volume of the trough is 5m * 1m * 4m = 20 cubic meters.
- The mass of the water is therefore 20 cubic meters * 1000 kg/m^3 = 20,000 kg.
- To raise this mass of water to a height of 3 meters, we need to calculate the gravitational potential energy using the formula: potential energy = mass * height * gravity, where gravity is 9.8 m/s^2.
- The potential energy required to raise the water to the top of the trough is therefore: 20,000 kg * 3 m * 9.8 m/s^2 = 588,000 Joules.
Now let's move on to the second part:
2. The work required to lift the water from the top of the trough to the outlet:
- The mass of the water at the top of the trough can be found by multiplying the cross-sectional area of the trough by its height.
- The cross-sectional area of the trough can be found by multiplying the base (1 meter) by the height (4 meters) and dividing it by 2, since it is an isosceles triangle.
- The cross-sectional area is therefore: (1m * 4m) / 2 = 2 square meters.
- The mass of the water at the top of the trough is therefore: 2 square meters * 1000 kg/m^3 = 2000 kg.
- To lift this mass of water to a height of 3 meters, we need to calculate the potential energy using the formula: potential energy = mass * height * gravity, where gravity is 9.8 m/s^2.
- The potential energy required to lift the water from the top of the trough to the outlet is therefore: 2000 kg * 3 m * 9.8 m/s^2 = 58,800 Joules.
To find the total work required to empty the trough, we add the work required for both parts:
Total work = Work to raise water to the top of the trough + Work to lift water from the top of the trough to the outlet
Total work = 588,000 Joules + 58,800 Joules
Total work = 646,800 Joules
Therefore, the amount of work required to empty the trough by pumping the water out of the outlet is 646,800 Joules.